Method for analyzing investments using overlapping periods

ABSTRACT

This invention consists of a process for the analysis and selection of financial investments based on a comparative analysis of performance and diversification. The method permits the manipulation of extremely large data sets in a manner that is simple to understand and convenient to use. This invention permits historical performance data for investments to be analyzed in respect of every possible investment period using any pre-existing or personally defined quantitative measurement algorithm. The user can apply his or her personal weightings to the various performance measurements based on a combination of attribute and time period to construct a customized scoring process, based on which a comparative ranking of the investments can be created. This invention also permits a complete universe of investments to be segmented into peer groups based on one of a number of similarity/dissimilarity criteria from which the user may choose.

TECHNICAL FIELD

The present invention relates generally to financial investmentanalysis, and, more specifically, to processes for selecting financialinvestments based on a comparative analysis of performance anddiversification.

BACKGROUND ART

The principal selection criteria for investments that will constitute aninvestment portfolio are performance and diversification.

Although there is no guarantee that past performance patterns will berepeated in the future, it is considered desirable to avoid investmentsthe historical performance of which has failed to meet some minimumcriteria or has been unstable or inconsistent.

In any market conditions we can expect that some investments willperform well and others will perform badly. The concept of riskdiversification is to construct a multi-investment portfolio so thatunder all market conditions some combination of good performers willalways offset the under-performers and the portfolio consistentlyachieves its objectives.

Performance

Quantitative performance data tends to begin by showing average returnbased on different variations of the underlying data, e.g., totalreturn, load-adjusted return or tax-adjusted return. The data may alsoinclude other standard performance measures such as volatility,semi-variance, drawdown, Sharpe ratio or Sortino ratio together withproprietary measures specific to the particular database provider.Perhaps the most widely recognized examples of the latter would be theStar Rating for mutual funds published by Chicago-based Morningstar,Inc. or the Timeliness Ranking for stocks published by New York-basedValue Line

In the case of mutual funds and other collective investment programmes,a second set of performance data is based on the performance attributionand style analysis approach favoured by institutional investors. Thegoal of performance attribution and style analysis is to divide a fundmanager's returns into two parts—style and skill. Style is the part ofthe returns that is attributable to market movements and is dominated bythe asset class mix in a portfolio. Skill is the part unique to themanager and is usually associated with individual security selectiondecisions within each asset class.

This analysis is usually accomplished through the construction ofregression-based models, an approach that has evolved from thepioneering work of William Sharpe who first developed the Capital AssetPricing Model. The models try to measure the systematic, causalrelationship between the price performance of a fund and the movement inone or more market indexes. The measure of a fund's systematicrelationship with a market index is called its ‘Beta’ while that portionof a fund's return that has no systematic relationship to the specifiedmarket indexes is called its ‘Alpha’. Although theoretically this is notcorrect, Alpha is often interpreted as representing the skill of themanager and used to rank manager performance.

Most analytical software today calculates the average performance of aninvestment over a specific term, e.g., the most recent 1, 3, or 5 years,selected calendar years, or since inception. In addition, manyanalytical tools compare investments by showing how much $10,000 wouldhave grown over a specific term. Most consumers believe that thesesimple averages and growth graphs reflect the results that would havebeen achieved for any shorter sub-period, or holding period, within thespecified term. However, analysis by the inventor shows that this isoften not the case, and the discrepancy can be very large. Thus, thereis a need for a better measurement that can capture not only performanceduring a single term but also the consistency of performance for allholding periods within that term.

The quantitative criteria commonly used to compare performance aremeasured in many different units and the range of values can verygreatly. For example, return and volatility are both measured inpercentages, but returns can be positive or negative whereas volatilitycan only be non-negative. In contrast, Sharpe ratio and correlation areboth measured in integers, but Sharpe ratio is unbounded, whereascorrelation must always take a value between −1 and 1. Typically, manysoftware applications for analyzing investments provide multiple fieldswith different performance measurements for comparison amonginvestments, but offer no methodology or technical capability to combinemultiple criteria into a single composite result. Where rankingcapability on single criteria is provided, the most common form ofranking is percentiles or simple ordinal rank. The limitation of thismeasurement is that it provides no information about the scale ofdifference in the relative performance of the ranked investments. Thus,there is a need for better investment analysis tools including a singlescore that allows easy comparison of investment performance.

Diversification

Mutual funds are most commonly grouped by applying a pre-definedclassification system to their underlying holdings. The classificationsystems are usually based on a combination of geography (US, Europe,Latin America, Pacific/Asia, Japan), sector (Communications, Financial,Health etc.) and style (large-cap, mid-cap, small-cap, value, growth,balanced) for equities and duration (long term, intermediate,short-term) or tax status (taxable, non-taxable) for bonds. ThusMorningstar Inc., mentioned above, defines four main groupings that arefurther subdivided into 48 categories. The Investment Funds StandardsCommittee of Canada defines five main grouping that are sub-divided into33 categories.

Under the style analysis approach, the simplest form of regression modelidentifies the single index with which the fund's performance is mostclosely related (this is sometimes referred to as the ‘best-fit index’)and funds can be grouped based on this criterion.

The investment strategies pursued by most mutual funds and ‘traditional’institutional investment management programmes are usually subject torestrictions on shorting securities or applying leverage and theinvestment manager is often constrained to buying and holding assets ina few well-defined asset classes. These buy-and-hold strategies lendthemselves to the two principal grouping methods described above.

In recent years however there has been an explosion of investment inhedge funds that employ considerably more sophisticated and dynamictrading strategies in pursuit of absolute returns with no systematicrelationship to the general market. These funds may employ a very widerange of techniques (including shorting and leverage), may trade in allmarkets (defined by asset type as well as geography) and use a diverserange of trading instruments (including futures, swaps, options andother financial derivative contracts).

Because time series of performance data is very limited, because thesefunds generally do not disclose detailed position information, andbecause of the dynamic nature and complexity of their tradingstrategies, traditional holdings-based or style analysis methods can notbe extended to these funds.

(Extensive efforts are being made to apply style analysis methods to theperformance of hedge funds but these efforts face many technicalproblems in the construction of appropriate indexes and as yet there areno generally accepted standards.)

A third grouping method has therefore been developed for this class offunds, based primarily on a description of the manager's strategy ratherthan the characteristics of the fund's holdings. Examples or of suchdescriptors are as follows: Long/Short Equity Hedge; Short-Only; EventDriven; Distressed Situations; Merger Arbitrage; Convertible Arbitrage;Fixed Income Arbitrage; Capital Structure Arbitrage; Credit Arbitrage;Mortgage-Backed Securities; Market Neutral; Relative Value; GlobalMacro; Emerging Markets; and Currency.

Many of these descriptors do not have standard definitions and manyfunds employ multiple strategies in multiple markets, making itdifficult to assign them to a single category. Therefore, although astrategy-labeling approach is widely used the resulting classificationsystems have not yet coalesced into a generally accepted common format.

DISCLOSURE OF THE INVENTION

This invention consists of methods that constitute a unique process forthe analysis of financial investments based on a comparative analysis ofperformance and diversification.

In this context, “investments” includes any financial asset or group offinancial assets in respect of which it is possible to trade based ongenerally accepted and regularly available periodic valuations andnon-tradable indices and benchmarks. Such investments may include, butare not limited to, individual securities (such as stocks or bonds),collective investment vehicles (such as mutual funds, closed-end funds,hedge funds, or commodities funds), specialist financial contracts(variable annuities or financial derivative contracts), real estate, orany combination thereof. However, in order to simplify the material wewill focus our discussion and examples primarily on mutual funds andfunds pursuing absolute return strategies (which we shall refer togenerically as “hedge funds”), although analogous issues arise withother investments.

This invention is unique in a number of respects, namely that:

The apparatus and methods permit the manipulation of extremely largedata sets in a manner that is simple to understand and convenient touse.

This invention permits historical performance data for investments to beanalyzed in respect of every possible investment period using anypre-existing or personally defined quantitative performance measurementalgorithm. (This process is hereinafter referred to as “Multi-PeriodAnalysis”).

The user can apply his or her personal weightings to the variousperformance measurements based on a combination of attribute and timeperiod to construct a customized utility function, based on which acomparative ranking of the Instruments can be created. (This process ishereinafter referred to as “Scoring”); and

This invention permits the complete universe of investments to besegmented into peer groups based on one of a number ofsimilarity/dissimilarity criteria from which the User may choose. (Thisprocess is hereinafter referred to as “Grouping”).

The invention is basically a method for analyzing the performance of aplurality of investments. The method includes: using a data source fromwhich can be derived the percentage increase or decrease in the value ofeach investment during each of consecutive reporting periods within agiven time frame; calculating values of an investment performancemeasurement for a plurality of overlapping holding periods within thetime frame, respectively; and using the resulting values to judge thedesirability of each investment.

The investments are each a tradable asset or a portfolio of tradableassets or a non-tradable index or benchmark.

Each reporting period is of the same standard length of time.

The investment performance measurement includes any quantitativemeasurement of the absolute performance of a single investment or anyquantitative measurement of its performance relative to that of anotherinvestment.

Each holding period is a period of time spanned by any combination ofconsecutive, contiguous reporting periods, such that the length of aholding period is a multiple of the standard length of the reportingperiod.

In another aspect, the method includes, for each investment, calculatinga weighted average of the values of the investment performancemeasurement and comparing the respective weighted averages of theinvestments.

The weighting factor to be applied to the value in respect of eachholding period may be selected by a user, but, in the absence of suchdetermination, by default shall be based on the length of the holdingperiod associated with each performance measurement value.

In another aspect, the method includes: calculating a weighted averageof the correlation between each pair of investments for a plurality ofholding periods; performing a mathematical conversion on the weightedaverage of correlation values such that these values are mapped into arange of positive values in which a higher positive value reflects agreater degree of negative correlation between the investments.; andusing such converted or mapped values to partition the investments intogroups such that the investments in each group are more highlycorrelated with each other than with those in any other group.

In another aspect, the method includes calculating the percentage of alldesignated holding periods in which the performance measurement for aninvestment was more desirable than a fixed reference value or that ofanother investment.

In another aspect, the invention includes calculating values of aplurality of performance measurements for the plurality of holdingperiods for each investment; calculating a weighted average of thevalues of the performance measurements; calculating in respect of eachweighted average its normalized value, which is the number of standarddeviations such weighted average lies above or below the mean of allweighted averages, for each performance measurement for the investments;for each performance measurement, calculating a weighted average of thenormalized values for each investment; and performing a mathematicalconversion on the resulting weighted averages such that the highestresulting weighted average is mapped to one-hundred percent, the lowestis mapped to zero percent and all other values are mapped within thisrange accordingly.

The weighting factor to be applied to each normalized value may beselected by the user but, in the absence of such determination, bydefault shall equal a fraction, the numerator of which equals the numberof performance measurements being averaged and the denominator of whichequals one hundred.

In another aspect, in respect of any performance measurement value wherea lower value is more desirable, the method includes multiplying thecorresponding stored normalized value by a factor of negative one priorto calculating a weighted average of the normalized values.

In another aspect, the method includes storing the values of theperformance measurement for each of the investments in a database priorto using the values to judge the desirability of each investment.

In another aspect, the method includes storing the weighted averages foreach of the investments in a database prior to using the values to judgethe desirability of each investment.

In another aspect, the method includes: calculating values of aplurality of performance measurements for the plurality of holdingperiods for each investment; for each investment, calculating thepercentage of all holding periods in which the performance measurementfor an investment was more desirable than a fixed reference value orthat of another investment; calculating a normalized value for eachpercentage outperformance value, wherein the normalized value is thenumber of standard deviations such percentage outperformance lies aboveor below the mean of all outperformance values, for each of theinvestments; for each performance measurement, calculating a weightedaverage of the normalized values for each investment; and performing amathematical conversion on the resulting weighted averages such that thehighest resulting weighted average is mapped to one-hundred percent, thelowest is mapped to zero percent and all other values are mapped withinthis range accordingly.

In another aspect, the method includes making an investment decisionbased on the results of the analysis.

In another aspect, the method includes calculating a probability of lossvalue by counting the number of the holding periods for which the returnwas negative and dividing the total by the number of the holdingperiods.

In another aspect, the method includes calculating the percentage ofholding periods in which the value of a designated performancemeasurement for one investment is more desirable than a designated fixedvalue or than the value of the same performance measurement for anotherinvestment

In another aspect, the performance measurement is a value representingthe return of each investment.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a table showing the performance of a fund and an index whenmeasured using traditional single periods and using the new multi-periodanalysis;

FIG. 2 is a table showing the returns on an investment for overlappingholding periods;

FIG. 3 is a diagram of a computer and databases for implementing themethod of the invention;

FIG. 4 is a table showing a scoring profile;

FIG. 5 is a table showing various scoring profiles for several groups;

FIG. 6 is a table showing an overall score for three investments for aselected weighting of performance measurements;

FIG. 7 is a table showing an overall score for three investments for aselected weighting of performance measurements;

FIG. 8 is a table showing correlations between a pair of investments;

FIG. 9 is a table showing weighted averages of correlations for multiplepairs of investments;

FIG. 10 is a simplified flow chart of a process of comparinginvestments;

FIG. 11 is a simplified flow chart showing a scoring process; and

FIG. 12 is a simplified flow chart showing a grouping process.

BEST MODE FOR CARRYING OUT THE INVENTION

Multi-Period Analysis

One of the biggest difficulties facing an investor who seeks to select anumber of mutual funds using currently available performance analyticsis that performance measurements are provided for only a limited numberof discrete periods.

Typically performance indicators such as return, volatility or sharperatio are calculated for periods of one, three, five, seven and tenyears, measured by calendar year or trailing from a recent month orquarter-end. In addition the period from inception to the present isoften included. Alpha and Bets calculations are similarly based on oneor perhaps two specific periods such as three or five years.

A major difficulty with this approach is that these average numbers canbe misleading and can lead to mistaken selection because they fail toadequately reflect the true performance history of a find. For example,Table 2 represents actual performance data for a US-based fund andcompares it to the performance of the S&P500 Index. This table showsthat the fund out-performed the S&P500 Index in each of the one-, three,five, seven- and ten-year periods ending September, 2001, which wouldseem to recommend it as a good candidate for investment. Using one ofthe leading mutual fund databases it was possible to identify 29 USdomestic mutual funds that outperformed the S&P500 Index in each ofthese periods and also for a fifteen-year period. Again this performancewould seem to recommend these funds for investment.

This invention however uses a different approach that provides greaterdepth and accuracy of analysis than is currently available. This is theMulti-Period Analysis Algorithm The method of this invention takes theperformance data in whatever frequency is available (in this casemonthly) between the dates in respect of which a comparison is required(October 1991 to September 2001 inclusive in the case of this example)and calculates the annualized return that would have been earned by aninvestor in every possible sub-period, or holding period, between thesedates. Thus there were 120 separate holding periods of one month each,119 holding periods of two months each, 118 holding periods of threemonths etc. down to two holding periods of 119 each and a single holdingperiod of 120 months. In this example, the total number of holdingperiods in respect of which the apparatus calculates returns is 7,260. Aweighted average of all these results is then calculated. The methodpermits the user to select their preferred weighting method. In thisexample, the weighting used equals the length of the relevant holdingperiod expressed in months. The result for the S&P500 Index, as shown inthe table of FIG. 1 is 17.46% and for the sample fund is 7.62%. Based onthese results, an investor might well decide not to invest in the fund.

The holding periods are called overlapping holding periods because, twoperiods of the same length, for example, the period spanning January,February, and March and the period spanning February, March and Aprilhave two months in common. Thus, the holding periods overlap.

How can we explain the sharply different results of using a few discreteperiods and using the comprehensive method incorporated into thisinvention? During the eighteen-month period ending September 2001 thefund outperformed the index by a spectacular 92.71% per annum. This wassufficient to ensure that the average fund performance, even spread overa ten-year period, exceeded the average for the index. In fact, however,the index actually outperformed the fund in 75% of all possibleinvestment holding periods (5,435 out of the total of 7,260). Thecurrent state-of-the-art methods accurately reported the average resultfor a few discrete holding periods but failed to reflect the fact thatmost of the fund's performance was concentrated within a specific veryshort space of time within these longer periods. This invention, byexamining all periods and applying a weighting ensures that suchshort-term aberrations do not dominate the comparative performanceanalysis.

Of course, in this extreme example, the extraordinary difference inperformance for the one-year period would give a strong indication to aninvestor that further research would be advised. However, with more than10,000 funds from which to choose, or even limited to the 29 fundsmentioned above, it becomes impractical if not impossible to manuallycarry out such a detailed comparison or to draw useful conclusions whenthe effect is not so extreme. The value of this invention is that themulti-period analysis directly solves this problem.

The same multi-period analysis can be applied to any quantitativemeasure. In addition, not only can the average of the multi-periodresults be compared across investments, the results for any twoinvestments may be compared for every single corresponding holdingperiod. This capability dramatically extends current fund performanceanalysis systems by providing a deeper, more detailed and totallycomprehensive analysis of a fund's historical performance.

In another example, the table of FIG. 2 shows a multi-period analysistable for an investment for which twelve months of data has beenanalyzed. FIG. 2 shows that a table of multi-period analysis results iswedge-shaped. In this example, the column labelled “Return” provides theactual, continuously compounded one-month return for the correspondingmonth. The values under the heading “Annualized, Continuously CompoundedReturn (%)” are the annualized, continuously compounded returns for thecorresponding holding periods, which are given along the horizontal axis(top row of the table).

Note that the return in any cell is the return for a period ending atthe end of the month indicated horizontally in the leftmost column. Thelength of the holding period associated with any return, as expressed inmonths in this example, is indicated vertically in the uppermost row.Thus, FIG. 2 shows that the annualized return for a five month holdingperiod ending in July was 24.54%. The annualized return for thefive-month holding period ending in August was 25.81%. Note that eachholding period is a period of time spanned by any combination ofconsecutive, contiguous reporting periods, such that the length of aholding period is a multiple of the standard length of the reportingperiod. In the example of FIG. 2, the standard reporting period lengthis one month, and each holding period is a multiple of one month. Ifinvesting data were reported at weekly intervals, that is, if thereporting period were one week, the length of each holding period wouldbe a multiple of one week, for example.

Investments may be analyzed simply by comparing the returns for the manyoverlapping holding periods. However, a preferred further step is todetermine the weighted average of the values in the table after formingthe multi-period return table. The weighting factor to be applied shallbe defined by the user; however, in the absence of such a definition,the weighting factor shall be defined as follows: The numerator of theweighting factor is the length of the particular holding period. Thedenominator of the weighting factor is the same for all holding periodsand equals the sum of a series of numbers. Each number is the product ofthe length of each holding period in months and the number of holdingperiods of that length. Using the table of FIG. 2 as an example, thereare twelve holding periods of one month, eleven two-month holdingperiods, ten three-month holding periods, and so on. The number ofholding periods of each length is given in the second row from the topin FIG. 2. Thus, the denominator of the weighting thus would be asfollows:(1×12)+(2×11)+(3×10)+(4×9)+ . . . +(9×4)+(10×3)+(11×2)+(12×1)=364

The weighted average multi-period return value for the investment of thetable of FIG. 2 is thus 25.1982%.

The weighted average provides a single measurement that captures thelevel, range and ordering of periodic returns over time. This permitsperiod-by-period comparative performance analysis among any combinationof investments, including single securities, portfolios, assets such asreal estate, or indices. Multi-period analysis is the only method bywhich one can answer the question “what would have been an investmentresult if one had randomly decided the holding period?” Multi-periodanalysis can test an investment manager's claim to have beaten themarket “over the past n years.” Further, the multi-period analysisprovides insights into whether a specific investment has performed bestover shorter or longer holding periods.

A similar table can be constructed with periodic return data for anyinvestment, including an index or benchmark, and a weighted averagemulti-period return value can be calculated in the same manner. Thus,the weighted average multi-period return value for the investment ofFIG. 2 can be compared to the other investments or indices or benchmarksto determine which is more desirable when compared on this basis.

Although the example of FIG. 2 uses return as the performancemeasurement, other return measurements such as volatility or SharpeRatio or any other quantitative performance measurement can be used aswell. That is, the information that is input to prepare the table isperiodic return data, but the entries in the table need not be returnvalues. The entries in the table can be other values such as volatilityor Sharpe Ratio or any other quantitative performance measurement.

Another way of judging investments based on the multi-period analysis isto calculate a probability of loss value for a given time frame. Theprobability of loss value is calculated by counting the number ofholding periods within the time frame for which the percentage return isnegative. This number is divided by the total number of overlappingholding periods within the time frame. The result is a probability ofloss value that is useful in judging the performance of investments. Ahigher probability of loss is less favourable as an indicator ofperformance than a lower probability of loss.

A further way of judging investments based on the multi-period analysisis to calculate a percentage outperformance value, which is thepercentage of holding period within a given time frame that aninvestment performance measurement for one investment was more desirablethan either a designated absolute value or than that of the sameperformance measurement for another designated investment or index. Thisis a more general application of the method used in calculatingprobability of loss.

More specifically, with reference to the table of FIG. 2, suppose onewished to know the percentage of holding periods in which the return ona particular investment was greater than 15%. There are a total of 78holding periods in the table of FIG. 2, and in 73 of those periods thereturn was greater than 15%. Thus, the percentage outperformance using15% as the criterion is 73/78×100 or 93.6%. A table like that of FIG. 2can be constructed for the S&P500 index for the same time period. Thenumber of holding periods in which a particular investment outperformedthe S&P500 index could be determined, and that number divided by 78would give a percentage outperformance using S&P500 index outperformanceas the criterion.

A further way of judging investments on the multi-period analysis is tocalculate a weighted average volatility value. The weighted averagevolatility value is the weighted average of numbers that represent thevolatility of an investment for every overlapping holding period withinthe given time frame. The result is useful for judging the volatility ofinvestments.

When the method is incorporated in software, which is the preferred wayof implementing the method, the software indicates to the user thecommon term for which data is available for all the investments the userwishes to compare. The user would then specify the desired start date,end date and minimum and maximum holding periods to be used.

Preferably, the multi-period analysis is performed by a computer usingsoftware that incorporates the method of the invention. Further, whenthe method is performed by a computer, the periodic investment data maybe taken from public database such as TASS (a hedge fund performancedatabase) or databases provided by the Center for Research in SecurityPrices (CRSP) or private databases. Tables like that of FIG. 2 can bepre-calculated for every investment within a source database before anyinvestment analysis is done. This will normally make analysis fasterwhen investments are selected for comparison or ranking.

One optional feature of the invention that speeds up analysis is thattables such as that of FIG. 2 are pre-calculated, as mentioned above,prior to any investment analysis. Many tables are pre-calculated andstored in a warehouse database, which is normally a local database butcan also be accessed over a local area network or over the internet. Forexample, each pre-calculated table represents one investment and oneperformance measurement and records the performance measurement forevery possible holding period between the earliest and the latest datesin respect of which return data is provided. Tables may be prepared forany quantitative measurement such as return, volatility, or Sharpe ratioand for correlations between pairs of investments. Thus, when there is aneed to compare two investments or to rank many investments, thesoftware need not calculate the performance measurement for each holdingperiod. The software program needs only to refer to the appropriatecells of the appropriate table in the database.

FIG. 3 shows a computer 10, which includes a display 12 and a user inputdevice 14. The computer 10 is connected to an investment historydatabase (or databases) 16. The investment history database (ordatabases) 16 can be stored locally, can be on portable media, such asCD ROM, or accessed over a local area network or over the Internet. Thecomputer 10 uses investment history data from the investment historydatabase (or databases) 16 to populate a warehouse database 20, whichincludes pre-calculated tables. The pre-calculated tables contain, forexample, among other things, multi-period return data for a universe ofinvestments. The warehouse database may be stored locally or it may beaccessed over a local area network or over the Internet. The computer 10runs software that performs the multi-period analysis described above onthe warehouse database. The user-interface of the computer 10, which isprogrammed to perform the method of this invention, indicates a commonterm for which investment data is available for all investments ofinterest The user may specify the desired start date, end date, andminimum and maximum holding periods. The user also may select theperformance measurement to be calculated from choices such as return,volatility, probability of loss, and Sharpe ratio. In addition, theinterface permits many other parameters to be set by the user. Theprogram provides numerical and graphical analysis of the results on thedisplay.

Alternatively, the computer need not use pre-calculated tables and neednot employ the warehouse database 20. All calculations can be done asneeded from the history database 16.

FIG. 10 is a self-explanatory flow chart showing stages of an exemplarymulti-period analysis in which investments are compared based onweighted averages and/or percentage of favourable holding periods. Thesteps of such a method depend on the user's goals and may be variedaccordingly. Step 30 is the step of calculating a value of an investmentperformance measurement for each holding period. Steps 34 and 32 neednot both be performed. Depending on the users goals, the user mayperform one or both of steps 32 and 34 or may simply use single valuesfor discrete holding periods from the values calculated in step 30. Step36 is a comparison step. For example, if only step 34 is performed andnot step 32, then in step 36, only the weighted averages would becompared to determine relative performance. If neither step 32 nor 34 isperformed, and the user instead chooses to use the performancemeasurement from a discrete holding period, then step 36 is simply astep of comparing the values calculated in step 30 for the chosenholding period. Step 38 is a step of making an investment decision basedon the comparison of step 36. For example, step 38 may include thepurchase of shares in a stock that compared favourably in step 36.

Scoring Process

Most investors will select funds based on a number of criteria and eachwill have his or her personal view as to the relative importance of eachcriterion to the final decision. The leading tools available todayprovide a vast range of performance measurements and an investor mayestablish a fund's rank ordering based on any single criterion. However,it appears that, until now, no method has been available by which aninvestor can freely combine multiple criteria to create a unified rankordering that reflects personal priorities. This invention provides justsuch functionality through scoring.

This feature of the method permits the user to specify the dates,between which the historical analysis will be applied, the range ofmultiple holding periods in respect of which the weighted performancemeasurement will be calculated (as described in the preceding section)and the criterion that will be applied to the selection process.Finally, the user specifies the relative importance of each selectioncriterion to the final decision. This may be expressed by a number ofmethods, including serial rank ordering or percentage weighting.

The method then includes producing a ranking for each fund in respect ofeach selection criterion. The ranking methodology is designed to make itindependent of the units in which each criterion is measured. The methodthen includes generating a scoring profile that specifies the manner inwhich the criteria are to be combined in accordance with the relativeimportance ascribed by the user. The result of applying the scoringprofile is to generate a single index with values between zero percentand one hundred percent and to assign an index value to each fund. Thecloser the index value is to one hundred percent the higher the rankingof the fund in terms of the user's personalized scoring process.

A different scoring profile can be defined for every group ofinvestments within the defined universe. Each scoring profile mightreflect, for example, the stated primary performance objectives of thegroup, e.g., high return or capital preservation.

For example, FIG. 4 shows a scoring profile table listing threeperformance measurements in the first column. The other columns show thestart date, the end date, the minimum holding period, the maximumholding period and the weight, which are chosen by the user. The weightrepresents the weight given to the corresponding performancemeasurement. One such profile can be selected for each of a plurality ofgroups of investments, as shown in the table of FIG. 5.

FIG. 5 shows a table listing three groups in the first column. In thesecond column, a scoring profile for the corresponding group is given.The scoring profile is simply the combination of the designatedperformance measurements to be used for scoring and the weight, orsubjective importance, as a percentage, given to each performancemeasurement. The sum of the weights must equal 100%.

The scoring process includes, for each group, calculating the raw valueof each performance measurement specified in the scoring profile. Then,the mean and the standard deviation of the raw values across the groupare calculated. In the case of the table of FIG. 4, the raw values arethe weighted averages of three different measurements of performance.However, each raw value may be a percentage outperformance value, whichwas described above. That is, a percentage outperformance value may beused for each of the performance measurements. Other raw values thatindicate performance may be selected by the user.

Then, for every investment, the scoring process includes counting thenumber of standard deviations the raw value is above or below thecorresponding mean. This is called the normalized value. Normalizedvalues have the statistical property that, irrespective of the units ormeasurement or the distribution of the underlying raw values, theirdistribution is “normal ” (in the statistical meaning) with a mean ofzero and a standard deviation of one.

For each investment, the user-specified weighting is applied to thenormalized value for each measurement and a weighted average iscalculated. This result is again normalized.

A score can be assigned in respect of a single criterion or to theweighted average of all criteria as follows. A score of 100% is assignedto the investment with the best normalized value within the group. Ascore if 0% is assigned to the investment with the worst normalizedvalue within the group. For all other investments, the assigned score isas follows: $1 - {\left\{ \frac{\quad\begin{matrix}{{BestNormalizedValue} -} \\{NormalizedValueofInvestmentbeingScored}\end{matrix}\quad}{{bestNormalizedValue} - {WorstNormalizedScore}} \right\} \times 100}$

FIG. 6 shows sample results from the scoring process. FIG. 6 is a tablefor three investments and two performance measurements in which returnis given a weight of 70% and probability of loss is given a weight of30%. In this table, the three investments, Fund A, Fund B, and Fund Care scored with a single score according to the weighting for the twoperformance measurements. Values that are in parentheses are negative.

FIG. 7 is a table like FIG. 6. However, in FIG. 7, the performancemeasurement of return is weighted at 30% and probability of loss isgiven a weight of 70%. The change in score that results in the change ofweighting can be seen in the rightmost column of the two tables, whichgives the overall score. The scoring process is preferably implementedwith a software program and performed by a computer 10, as describedwith reference to the multi-period analysis. The weightings and otheruser-selected variables are entered by a user using the user inputdevice 14, when prompted by the user interface.

FIG. 11 is a self-explanatory flowchart showing an exemplary procedurefor scoring. In the procedure of FIG. 11, the scoring is based onweighted averages of a performance measurement or the percentage offavourable holding periods for a particular performance measurement. Thesteps of the scoring process will vary according to the user's goals butmay be as shown in FIG. 11. Referring to FIG. 11, step 40 is the step ofcalculating investment performance measurements for each holding periodEither or both of steps 44 and 42 may be performed or the user may godirectly to step 46 by choosing to use values calculated in step 40 fordiscrete holding periods. Step 44 is a step of calculating weightedaverages of the investment performance measurements over the holdingperiods. Step 42 involves calculating a percentage outperformance, asdescribed above. If only step 44 were performed and not step 42, forexample, then in step 46, only the weighted averages would be used incalculating the number of standard deviations. If neither step 42 norstep 44 is performed, the user may simply in step 46 calculate thenumber of standard deviations based on the performance measurementvalues for a selected discrete holding period. In step 48, a weightedaverage of the standard deviation values is calculated for eachinvestment and for each performance measurement. In step 50, theresulting weighted averages are mapped between zero and one hundred.

Grouping

Grouping methodologies that use portfolio holdings or strategy labelssuffer from a number of weaknesses, including that they are not directlybased on actual performance. Regression-based style analysis of courseuses performance data but share other label-based shortcomings. Inaddition, many individual investors are less concerned about how closelytheir fund tracks an index than there are in achieving a fixed returnobjective, usually related to their life plan, such as fundingchildren's education or providing for retirement. In addition, none ofthese state-of-the art approaches has yet been applied successfullyacross both mutual funds and hedge funds.

A common characteristic of current methods is that the number of groupsinto which the funds may be divided is fixed by the methodology and isnot related directly to how many funds the investor wishes to select.This is a critical issue. For example, let us assume that one of theexisting systems produces twenty-six categories. If an investor wants toselect exactly twenty-six funds, then he or she might decide thatpicking one fund from each category will provide the highest degree ofdiversification. If however the investor wishes to invest in only twelvefunds, ideally he or she would prefer to be able to reorganize all ofthe funds into just twelve groups with the highest possiblediversification and again pick one from each group. Current methods haveno algorithms to achieve such a regrouping.

This invention directly solves the problem using a performance-basedpeer grouping process. The preferred performance measurement used inthis process is correlation, although the invention supports the use ofother measurements. Correlation is a statistical technique that can showwhether and how strongly pairs of variables are related. The sign of thecorrelation coefficient, which can be either positive or negative,defines the direction of the relationship. A positive correlationcoefficient means that as the value of one variable increases, the valueof the other variable increases; as one decreases the other decreases. Anegative correlation coefficient indicates that as one variableincreases, the other decreases, and vice-versa. Combining investmentsthat have a negative correlation to each other is usually expected toproduce a more stable return across different market environments.

It is relatively easy to work with correlation for small data sets.Therefore once the complete universe has been reduced to a relativelysmall number of finds selected for the final portfolio, correlation isan important element in all established methods for deciding whatpercentage of capital should be allocated to each investment—oftenreferred to as portfolio optimization”.

Because correlation is calculated directly from performance data andrepresents the systematic relationship among the investments, it is oneof the best possible criteria for creating peer groups for riskdiversification. However, there are a number of significant technicalchallenges in working with correlation data for a very large universe offunds. For example, a universe of 10,000 funds would create close to 50million different pair-wise correlation coefficients so the scale ofdata alone might discourage investigation in this area.

This invention uses advanced partitioning techniques in a multi-stageprocess that can group any large universe of investments based on theirpair-wise correlation or on other user-specified measures ofsimilarity/dissimilarity based, for example on the absolute differencein returns between each pair of investments either in respect of aplurality of single performance reporting periods or in respect of aplurality of designated holding periods. The user may specify the numberof groups into which the universe should be divided and this of coursemay be determined by the number of investments to be included in theportfolio. In addition, of course, the correlation coefficients or othermeasures may be calculated using the Multi-Period Analysis describedabove, or by any other method.

The grouping process implements a clustering methodology called“Partitioning Around Mediods” as detailed in chapter 2 of L. Kaufman andP. J. Rousseeuw, Finding Groups in Data: An Introduction to ClusterAnalysis, Wiley, New York (1990).

In the preferred embodiment, a universe of investments is partitionedinto a user-specified number of groups based on the correlation ofhistorical performance between each pair of investments within theinvestment universe or other measurement of similarity/dissimilarity.The user can specify the number of groups. The correlation inputs arecalculated using the multi-period analysis described above. The processcan be applied to any combination of investments (stocks, bonds, mutualfunds, hedge funds, indices, or benchmarks).

The process also allows the user to partition the investment universeusing the data provider's labelling system and to compare the groupingresults obtained from applying different grouping methodologies.

First, common start and end dates for all investments within theinvestment universe are determined. Then, for every pair of investments,the weighted average of the correlations over the overlapping holdingperiods within the common term is calculated. The holding periods aredefined in the same manner as described above with respect to themulti-period analysis. Using the correlation values, the investmentuniverse is divided into a specified number of groups such that theinvestments in each group are more highly correlated with each otherthan with those in any other group.

To divide the investment universe into the specified number of groups,dissimilarity values are employed. Each dissimilarity value is a singlenumber that measures the degree of similarity or dissimilarity betweentwo objects in the dataset. The lower the dissimilarity value, the moresimilar the two objects are, the higher the dissimilarity value, themore dissimilar the two objects are.

For each pair of investments being considered, that is, for each pair inthe investment universe, the correlations are determined for designatedmultiple holding periods. If specific holding periods are notdesignated, the default is to begin with all holding periods of at leasttwo reporting periods in length and end with the holding period thelength of which equals the length of the common term. The table of FIG.8 is an example of a table of such correlations for two investments,investment B and investment E. Because of the way correlation iscalculated, the correlations will always lie between negative one andone. Therefore, the values in the cells of the table of FIG. 8 willalways lie between negative one and one. The weighted average of thecorrelations of the table of FIG. 8 is 0.4849919, as indicated near thetop of the table.

A weighted average of these correlations is calculated for eachinvestment pair in the investment universe. The weighting factor can beselected by the user, but the default weighting factor is based on thelength of each holding period, as described with respect to multi-periodanalysis above. The result can be arranged and displayed in a table likethat of FIG. 9, which may be referred to as a pair correlation table,since it shows the correlations between pairs of investments. The termof the table of FIG. 9 is twelve months. Although most of thecorrelations in the pair correlation table of FIG. 9 are represented bythe subscripted variable “Corr,” the numerical value of the correlationbetween investments B and E, which is derived from the correlation tableof FIG. 8, has been written in the pair table in the appropriate cell.In the pair correlation table of FIG. 9, Corr_(xy) refers to thecorrelation between investment X and investment Y. The cell forCorr_(eb) shows the correlation value calculated as the weighted averageof correlation values for all holding periods of 2 months or longerduring the 12-month term in respect of investment E and investment Baccording to the correlation table of FIG. 8. The table also shows thatevery investment's correlation with itself is one.

In this context, a correlation value of positive one indicates leastdissimilarity, whereas a correlation value of negative one indicatesgreatest dissimilarity. The preferred process of partitioning aroundmedoids (the process described by Kaufman and Rousseeuw mentionedabove), however, is designed to work with positive values where highervalues indicate greater dissimilarity. Therefore, after taking theweighted average, the correlations are converted to positive values. Thepositive values are the dissimilarity values mentioned above. In otherwords, a mathematical conversion is performed on the correlation valuessuch that negative values are mapped to positive values. In thepreferred embodiment, a correlation value of negative one is mapped totwo, a correlation value of one is mapped to zero, and all othercorrelation values are mapped within a range from zero to twoaccordingly. This is achieved by subtracting each value from one togenerate a corresponding dissimilarity value; however, other similarconversions that produce positive numbers, such that a higher positivenumber denotes greater dissimilarity can be used.

The resulting dissimilarity values are used by software thatincorporates Kaufmnan and Rousseeuw's partitioning method to group theinvestments into a specified number of groups such that the investmentsin each group are more highly correlated with each other than with thosein any other group. Although the “Partitioning Around Medoids” methoddescribed by Kaufman and Rousseeuw is presently preferred, other methodsthat can partition the groups such that the investments in each groupare more highly correlated with each other than with those in any othergroup may be used.

The resulting groups can be used to improve risk diversification. Thatis, the groups can be used to construct a portfolio of investments inwhich an investor may have greater confidence that under all marketconditions, some combinations of good performers will likely offset theunder-performers, and the portfolio will consistently achieve itsobjectives.

FIG. 12 is a self-explanatory flow chart showing an exemplary groupingprocess. The steps will vary according to the user's goals but may be asshown in FIG. 12. Referring to FIG. 12, step 60 is the step ofcalculating the correlation between their returns for each pair ofinvestments for each holding period within a given time frame. In step62, the weighted average of the multi-period correlations between eachpair of investments is calculated. The correlation values are convertedto positive values such that higher positive numbers indicate greaterdissimilarity in step 64. In step 66, the converted values are used topartition the investments into groups such that the investments in eachgroup are more highly correlated with each other than with those in anyother group. Step 68 involves choosing a portfolio of investments inwhich risk is diversified using the groups. More specifically, one wouldnormally not choose investments that are all in the same group ifdiversity is a goal. Ideally, the portfolio would include investmentsfrom more than one group for diversification.

The grouping process is preferably implemented with a software programand performed by a computer 10, as described with reference to themulti-period analysis. The number of groups and other user-selectedvariables are entered by a user using the user input device 14, whenprompted by the user interface.

A software process has been developed for implementing the Kaufman andRousseeuw method of portioning. That process is referred to as the PAM(Partitioning Around Mediods) algorithm. The following is a detaileddescription of the PAM algorithm:

The PAM Algorithm has two stages, the Build Stage and the Swap Stage.

The purpose of the Build Stage is to identify a first set of Medoidsequal to the desired number of groups.

In the Swap Stage all non-Medoid Objects are iteratively tested to seeif they are better qualified than the existing selected Medoids.Usually, after each iteration of the process, one Candidate is selectedto replace one existing Medoid. The process stops when no betterqualified Candidate exists.

Glossary

A Candidate is an Object that has not yet been selected as a Medoid.

An Object is one of the members of the dataset being partitioned

A Test Object is the name given in the Swap stage to each Object in turnagainst which the swap test is applied.

DValue refers to the dissimilarity value mentioned earlier. It is asingle number that measures the degree of similarity or dissimilaritybetween two Objects. The lower the DValue, the more similar the twoObjects are. The higher the DValue, the more dissimilar the two Objectsare.

DValue (Object a×Object b) refers to the DValue for the indicated pairof Objects.

DVector refers to the Dissimilarity Vector for a single Object. Itcontains the same number of elements, as there are Objects in thedataset and shows the respective Object's DValue compared to all otherObjects (including itself).

Dz and DzSky are counters used in the Swap Stage (see below).

HDValue means the highest DValue across all of the DVectors for all ofthe Objects in the dataset.

A Medoid is the Object in a group that has the greatest similarity toall other Objects in the group. This means that the aggregate DValuesfor (Medoid×Each other Object in the group) is the lowest among groupmembers.

RValues mean the values used in the RVector.

RVector refers to the Reference Vector that is used in the Build Stageto find Medoids. This vector has the same number of elements as thereare Objects in the dataset, which is also the same number of elements inany Object's DVector.

Vector A is used in the Swap Stage. For each Object, the Medoid forwhich DValue (Object×Medoid) is lowest, i.e., the Medoid to which eachObject is most similar, is identified. This is called the Object'sMedoid A. Vector A consists of the DValues for each Object with itsrespective Medoid A.

Vector B is also used in the Swap Stage. For each Object, identify theMedoid for which DValue (Object×Medoid) is second lowest. This is calledthe Object's Medoid B. Vector B consists of the DValues for each Objectwith its respective Medoid B.

Build Stage

The purpose of the Build Stage is to identify the first group ofMedoids. The number of Medoids will equal the final number of groupsdetermined by the user.

To find the First Medoid:

Step 1: Create the first RVector. In the case of the first RVector, eachelement is given the same RValue which is arbitrarily calculated by theformula IRValue=(HDValue*1.1)+1.

Step 2: Select a Candidate and, one-by-one, subtract each DValue in theCandidates's DVector from the corresponding RValue in the RVector andsum all of the results.

Step 3: Repeat Step 2 for all Candidates. The Candidate with the highestaccumulated total is selected as the first Medoid.

To find Subsequent Medoids:

Step 4: Calculate a new RVector. To construct the RVector to findsubsequent Medoids, base the RVector upon the RVector used to identifythe previous Medoid. For each Object in the Dataset, enter the lower ofDValue (Object×Most Recently Identified Medoid) and the correspondingprevious RValue within the prior iteration's RVector

Step 5: Select a Candidate and one-by-one, we subtract each DValue inthe Candidate's DVector from the corresponding RValue in the RVectorselected in Step 4.

In respect of DValue (Candidate×Object b) the “corresponding” RValuewould be the lower of DValue (Object b×Most Recently Identified Medoid)and DValue (Object b×Medoid identified one iteration before MostRecently Identified Medoid).

Sum only those differences that are positive. (Note: in the case offinding the First Medoid, by construction all the differences arepositive so all are summed as stated in Step 2 above)

Step 6: Repeat Step 5 for all Candidates. The Candidate with the highestaccumulated total is selected as the next Medoid.

Step 7: Repeat Steps 4-6 until we have identified the same number ofMedoids (including the first Medoid) as the desired number of groups.

Swap Stage

Step 8: Construct Vectors A and B according to the Glossary description.

Step 9: Set DzSky=1

Step 10: Set Dz=0

Step 11: Select the First Candidate , the first Medoid and the firstTest Object. Calculate:

D1=DValue (First Medoid×Test Object); and

D2=DValue (Candidate×Test Object)

Step 12: If D1=DValue (Test Object's Medoid A×Test Object),

CalculateMin [DValue (Test Object's Medoid B×Test Object), D2]−DValue (TestObject's Medoid A×Test Object)

Accumulate to Dz

Else

If D2<DValue (Test Object's Medoid A×Test Object)

Calculate D2 −DValue (Test Object's Medoid A×Test Object)

Accumulate to Dz

Else

Do not accumulate.

Step 13: Repeat Steps 9 and 10 for all Test Objects, without changingthe First Candidate or the First Medoid.

The grand total is the Dz value for the First Candidate and the FirstMedoid

Step 14: If Dz<Dz Sky, then set DzSky=Dz and make a note of whichCandidate and which Medoid it happens for.

Step 15: Repeat Steps 10-14 for the First Candidate and each Medoid, andthereafter for each Candidate and each Medoid combination, until everyCandidate, Medoid, Test Object combination has been used.

Step 16: Finally we know which Candidate and Medoid pairing had thelowest Dz value so

If this lowest Dz (the last DzSky value)<0 then replace Medoid with thatCandidate.

Else

Don't replace.

Step 17: Repeate Steps 8-16 until no replacement is made (i.e., the lastDzSky value is not<0).

This is one example of a computerized process for partitioning. Asstated earlier, any process that can partition the groups such that theinvestments in each group are more highly correlated with each otherthan with those in any other group may be used

While the above description is of the preferred embodiment of thepresent invention, it should be appreciated that the invention may bemodified, altered, or varied without deviating from the scope and fairmeaning of the following claims. For example, tables of pre-calculatedperformance returns or other performance measurements are preferablyused when analyzing investments; however, the performance measurementsmay be calculated only when needed, without the use of pre-calculatedtables. In addition, the various multi-period values and the varioussteps in the processes described above may be may be used in differentsequences or may be used in part but not in whole depending on theuser's specific requirements.

1. A method for analyzing the performance of a plurality of investments,the method comprising: using a data source from which can be derived thepercentage increase or decrease in the value of each investment duringeach of consecutive reporting periods within a given time frame;calculating values of an investment performance measurement for aplurality of overlapping holding periods within the time frame,respectively; using the resulting values to judge the desirability ofeach investment.
 2. The method of claim 1, wherein the investments areeach a tradable asset or a portfolio of tradable assets or anon-tradable index or benchmark.
 3. The method of claim 1, wherein eachreporting period is of the same standard length of time.
 4. The methodof claim 1, wherein the investment performance measurement includes anyquantitative measurement of the absolute performance of a singleinvestment or any quantitative measurement of its performance relativeto that of another investment.
 5. The method of claim 1, wherein eachholding period is a period of time spanned by any combination ofconsecutive, contiguous reporting periods, such that the length of aholding period is a multiple of the standard length of the reportingperiod.
 6. The method of claim 1, wherein the method includes, for eachinvestment, calculating a weighted average of the values of theinvestment performance measurement and comparing the respective weightedaverages of the investments.
 7. The method of claim 6, wherein theweighting factor to be applied to the value in respect of each holdingperiod may be selected by a user, but, in the absence of suchdetermination, by default shall be based on the length of the holdingperiod associated with each performance measurement value.
 8. The methodof claim 1 wherein the method includes: calculating a weighted averageof the correlation between each pair of investments for a plurality ofholding periods; performing a mathematical conversion on the weightedaverage of correlation values such that these values are mapped into arange of positive values in which a higher positive value reflects agreater degree of negative correlation between the investments.; andusing such converted or mapped values to partition the investments intogroups such that the investments in each group are more highlycorrelated with each other than with those in any other group.
 9. Themethod of claim 1, wherein the method includes calculating thepercentage of all holding periods in which the performance measurementfor an investment was more desirable than a fixed reference value orthat of another investment.
 10. The method of claim 1, wherein themethod includes: calculating values of a plurality of performancemeasurements for the plurality of holding periods for each investment;calculating a weighted average of the values of the performancemeasurements; calculating in respect of each weighted average itsnormalized value, which is the number of standard deviations suchweighted average lies above or below the mean of all weighted averages,for each performance measurement for the investments; for eachperformance measurement, calculating a weighted average of thenormalized values for each investment; and performing a mathematicalconversion on the resulting weighted averages such that the highestresulting weighted average is mapped to one-hundred percent, the lowestis mapped to zero percent and all other values are mapped within thisrange accordingly.
 11. The method of claim 10, wherein the weightingfactor to be applied to each normalized value may be selected by theuser but, in the absence of such determination, by default shall equal afraction, the numerator of which equals the number of normalized valuesbeing averaged and the denominator of which equals one hundred.
 12. Themethod of claim 10, wherein the method includes, in respect of anyperformance measurement value where a lower value is more desirable,multiplying the corresponding stored normalized value by a factor ofnegative one prior to calculating a weighted average of the normalizedvalues.
 13. The method of claim 1, wherein the method includes storingthe values of the performance measurement for each of the investments ina database prior to using the values to judge the desirability of eachinvestment.
 14. The method of claim 6, wherein the method includesstoring the weighted averages for each of the investments in a databaseprior to using the values to judge the desirability of each investment.15. The method of claim 1, wherein the method includes: calculatingvalues of a plurality of performance measurements for the plurality ofholding periods for each investment; for each investment, calculating apercentage outperformance value, which is the percentage of all theholding periods in which each performance measurement was more desirablethan either a fixed reference value or that of another investment;calculating a normalized value for each percentage outperformance value,wherein the normalized value is the number of standard deviations suchpercentage outperformance lies above or below the mean of alloutperformance values, for each of the investments; for each performancemeasurement, calculating a weighted average of the normalized values foreach investment; and performing a mathematical conversion on theresulting weighted averages such that the highest resulting weightedaverage is mapped to one-hundred percent, the lowest is mapped to zeropercent and all other values are mapped within this range accordingly.16. The method of claim 15, wherein the weighting factor to be appliedto each normalized value may be selected by the user but, in the absenceof such determination, by default shall equal a fraction, the numeratorof which equals the number of performance measurements being averagedand the denominator of which equals one hundred.
 17. The method of claim15, wherein the method includes, in respect of any performancemeasurement value where a lower value is more desirable, multiplying thecorresponding normalized value by a factor of negative one prior tocalculating a weighted average of the normalized values.
 18. The methodaccording to claim 1, wherein the method includes making an investmentdecision based on the results of the analysis.
 19. The method accordingto claim 1, wherein the method includes calculating a probability ofloss value by counting the number of the holding periods for which thereturn was negative and dividing the total by the number of the holdingperiods.
 20. The method according to claim 1, wherein the methodincludes calculating the percentage of a designated set of holdingperiods in which the value of a designated performance measurement forone investment is more desirable than a designated fixed value or thanthe value of the same performance measurement for another investment